I feel like I mention distributivity a lot. And that’s fine with me! But I haven’t mentioned that distributivity ought to be a functor. Why not, eh? Doubtless there is some fancy and slick (*sicut oleo in barba Aaron…*) argument for why constructions that are natural enough naturally become functors… but that provocative link there isn’t quite it, I don’t think. Anways, yes, Distributivity seems to mean **not only** that colimits of pullbacks of things over given colimits are the original things themselves, **but also** pullbacks of commuting square over a map-of-colimits have the right map-of-colimits… a picture will help. Fix a diagram $D$, functors $F,G : D \to \mathcal{Top}$ and transformation $F \to G$

\[ \xymatrix{

{f^* F} \ar[rrr] \ar[ddd] \ar[dr] & & & {g^* G} \ar[ddd] \ar[dl] \\

& F(D) \ar[r] \ar[d] & G(D) \ar[d] \\

& \colim F \ar[r]|{!} & \colim G \\

U \ar[rrr]_w \ar[ur]_f & & & V \ar[ul]_g } \] in this diagram, distributivity asserts that $U = \colim f^* F$, $V = \colim g^* G$, and that $ w $ is the map-of-colimits induced by the natural transformation at top. (what else *could* the map-of-colimits be, anyway? It’s given universality of a colimit!)